On the exceptional set for binary Egyptian fractions

TL;DR

This paper investigates the set of integers for which a specific binary Egyptian fraction equation has no solutions, providing an improved asymptotic formula and analyzing the underlying group structure.

Contribution

It derives a precise asymptotic formula for the exceptional set in binary Egyptian fractions, improving previous results significantly.

Findings

Asymptotic formula for the exceptional set $E_a(N)$ established

Analysis of the underlying group structure used in proof

Improvement over previous bounds in the literature

Abstract

For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.